Galaxy star formation rates (SFR), Hi masses, and H2 masses all show strong correlations with stellar mass for star-forming systems. Moreover, galaxies with high/low SFR also have high/low gas contents. Here we compare observations of correlated deviations off the main sequence from Janowiecki et al. to those in three galaxy formation simulations: Simba, EAGLE, and IllustrisTNG. Simulations predict that H2 deviations are linearly correlated with SFR deviations, as expected since star formation directly arises in dense gas. However, Hi deviations predictions vary, with Simba producing a near-linear slope but EAGLE and IllustrisTNG predicting sub-linear slopes. At earlier cosmic epochs, a sub-linear slope is always present in EAGLE and IllustrisTNG, but only at z = 2 in Simba, becoming linear at z ≤ 1. This suggests that the linear slope owes to the onset of AGN feedback, and hence that such feedback must remove Hi from the galaxy to match the observed linear slope. This provides unique insights into baryon cycling processes that connect gas within galaxies and their surroundings.